Finite Group Databasebeta

← #326595

Subgroups — #326595

Order 512.

459 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22310C211yesmaximal, characteristic, complemented
2#22282C211yesmaximal, characteristic
3#10665C211yesmaximal, characteristic, complemented
4#10665C211yesmaximal, characteristic, complemented
5#10370C211yesmaximal, characteristic, complemented
6#10370C211yesmaximal, characteristic, complemented
7#9753C211yesmaximal, characteristic
8#9750C211yesmaximal, characteristic
9#9632C211yesmaximal, characteristic, complemented
10#9629C211yesmaximal, characteristic, complemented
11#8880C211yesmaximal, characteristic, complemented
12#8880C211yesmaximal, characteristic, complemented
13#8878C211yesmaximal, characteristic, complemented
14#8878C211yesmaximal, characteristic, complemented
15#8782C211yesmaximal, characteristic, complemented
16#3147C2211yescharacteristic
17#2932C2211yescharacteristic
18#2932C2211yescharacteristic
19#2399C2211yescharacteristic
20#1985—21no—
21#1984—21no—
22#1963C2211yescharacteristic
23#1963C2211yescharacteristic
24#1940C2211yescharacteristic
25#1940C2211yescharacteristic
26#1917—21no—
27#1916—21no—
28#1876—21no—
29#1876—21no—
30#1876—21no—
31#1876—21no—
32#1837C2211yescharacteristic
33#1832C2211yescharacteristic
34#1770C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36#1746C2211yescharacteristic
37#1571C2211yescharacteristic
38#1571C2211yescharacteristic
39#1570C2211yescharacteristic
40#1570C2211yescharacteristic
41#1564C2211yescharacteristic
42#1564C2211yescharacteristic
43#1502C2211yescharacteristic
44#1502C2211yescharacteristic
45#1502C2211yescharacteristic
46#1502C2211yescharacteristic
47#1480C2211yescharacteristic
48#1480C2211yescharacteristic
49#1480C2211yescharacteristic
50#1480C2211yescharacteristic
51#1303C2211yescharacteristic
52#1303C2211yescharacteristic
53#1303C2211yescharacteristic
54#1303C2211yescharacteristic
55#1281C2211yescharacteristic
56#1281C2211yescharacteristic
57#1274C2211yescharacteristic
58#1274C2211yescharacteristic
59C2×#135—21no—
60C2×#135—21no—
61C2×#131—21no—
62C2×#131—21no—
63#532C2311yescharacteristic
64#532C2311yescharacteristic
65C2×C4×Q8—21no—
66#495—21no—
67#495—21no—
68#494Q811yescharacteristic, complemented
69#493Q811yescharacteristic, complemented
70#492Q811yescharacteristic, complemented
71#492Q811yescharacteristic, complemented
72#435—41no—
73#435—41no—
74#435—41no—
75#435—41no—
76#431C2311yescharacteristic
77#431C2311yescharacteristic
78#431C2311yescharacteristic
79#431C2311yescharacteristic
80#423C2311yescharacteristic
81#423C2311yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×#104—21no—
87C2×#104—21no—
88C2×#104—21no—
89C2×#104—21no—
90#408—21no—
91#408—21no—
92C2×#97—21no—
93C2×#97C2311yescharacteristic
94C2×C4×C8—21no—
95C2×C4×C8C2311yescharacteristic
96#398—21no—
97#398—21no—
98#395—21no—
99#395—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4C2311yescharacteristic
102C4×C4⋊C4C2311yescharacteristic
103#376C2311yescharacteristic
104#341—21no—
105#341—21no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336D411yescharacteristic
113#336D411yescharacteristic
114#336D411yescharacteristic
115#336D411yescharacteristic
116#335—21no—
117#335—21no—
118#335—41no—
119#335—41no—
120#334—21no—
121#334—21no—
122#334—41no—
123#334—41no—
124#327—21no—
125#327—21no—
126#327—21no—
127#327—21no—
128#327—21no—
129#327—21no—
130#327—21no—
131#327—21no—
132D4⋊C8—41no—
133D4⋊C8—41no—
134D4⋊C8—41no—
135D4⋊C8—41no—
136C2×#41—21no—
137C2×#41—21no—
138C22×Q8—21no—
139#135—81no—
140#135—81no—
141#135—81no—
142#135—81no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×SD16—21no—
146C2×SD16—21no—
147C2×SD16—21no—
148C2×SD16—21no—
149C2×D8—21no—
150C2×D8—21no—
151#131—81no—
152#131—81no—
153#131—81no—
154#131—81no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—41no—
158C2×M16—41no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—41no—
162C22×C8—41no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8C2×Q811yescharacteristic
168C4⋊Q8C2×Q811yescharacteristic
169#127C2×Q811yescharacteristic
170#127C2×Q811yescharacteristic
171#124—21no—
172#124—21no—
173#121—21no—
174#121—21no—
175#121—21no—
176#121—21no—
177C4×Q8—81no—
178C4×Q8—81no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2411yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—81no—
208C4⋊C8—81no—
209#104—81no—
210#104—81no—
211#104—81no—
212#104—81no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—42no—
218C22⋊C8—42no—
219#97—21no—
220#97—21no—
221#97—81no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—81no—
227C4×C8C2×Q811yescharacteristic
228C4×C8C2×Q811yescharacteristic
229C4×C8#4111yescharacteristic
230C4×C8#4111yescharacteristic
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95#4111yescharacteristic
236#95#4111yescharacteristic
237#95#4111yescharacteristic
238#95#4111yescharacteristic
239#41—41no—
240#41—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246#41—41no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—42no—
250C2×Q8—81no—
251C2×Q8—81no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4111yescharacteristic
271Q16—81no—
272Q16—81no—
273SD16—81no—
274SD16—81no—
275SD16—81no—
276SD16—81no—
277D8—81no—
278D8—81no—
279M16—41no—
280M16—41no—
281M16—41no—
282M16—41no—
283M16—82no—
284M16—82no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—82no—
308C2×C8—82no—
309C2×C8C22⋊Q811yescharacteristic
310C2×C8C22⋊Q811yescharacteristic
311C2×C8C22⋊Q811yescharacteristic
312C2×C8C22⋊Q811yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—81no—
320C4⋊C4—81no—
321C4⋊C4—81no—
322C4⋊C4—81no—
323C4⋊C4—81no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42C22×Q811yescharacteristic
337C42C2×#4111yescharacteristic
338C42H(5,𝔽2)11yescharacteristic
339C42H(5,𝔽2)11yescharacteristic
340C23—41no—
341C23—41no—
342C23#52511yescharacteristic
343Q8—41no—
344Q8—41no—
345Q8—41no—
346Q8—41no—
347Q8—84no—
348D4—41no—
349D4—41no—
350D4—41no—
351D4—41no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4C2×C22⋊Q811yescharacteristic
387C2×C4#53511yescharacteristic
388C2×C4#55411yescharacteristic
389C2×C4#55411yescharacteristic
390C2×C4#55911yescharacteristic
391C2×C4#55911yescharacteristic
392C2×C4#57611yescharacteristic
393C2×C4#57611yescharacteristic
394C2×C4#57711yescharacteristic
395C2×C4#57711yescharacteristic
396C8—21no—
397C8—21no—
398C8—21no—
399C8—21no—
400C8—41no—
401C8—41no—
402C8—81no—
403C8—81no—
404C8—81no—
405C8—81no—
406C8—81no—
407C8—81no—
408C8—81no—
409C8—81no—
410C22—21no—
411C22—21no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#258011yescharacteristic
419C22#312211yescharacteristic
420C22#312611yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—42no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4#308111yescharacteristic
434C4#308111yescharacteristic
435C4#322411yescharacteristic
436C4#322411yescharacteristic
437C2—21no—
438C2—21no—
439C2—161no—
440C2—161no—
441C2#2926311yescharacteristic
442C2#2927011yescharacteristic
443C2#2927111yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2580
prime radical#3147C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22310C2semidirect
3. #10665C2semidirect
4. #10665C2semidirect
5. #10370C2semidirect
6. #10370C2semidirect
9. #9632C2semidirect
10. #9629C2semidirect
11. #8880C2semidirect
12. #8880C2semidirect
13. #8878C2semidirect
14. #8878C2semidirect
15. #8782C2semidirect
68. #494Q8semidirect
69. #493Q8semidirect
70. #492Q8semidirect
71. #492Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#223102
#222822
#106652
#103702
#97532
#97502
#96322
#96292
#88802
#88782
#87822